SMM306 Advanced Stochastics Coursework, ©2021 Alesˇ Cˇerny´ 1
Groups: Groups of max 3 students; self-selected
Deadline: Monday, 22/03/2021, details on Moodle
Questions: Only in live sessions, in front of the whole class
Submission guidelines: Typeset report of no more than 5 pages. Make sure the reported
results can be easily replicated using the submitted code. Use appropriate references but
also provide a self-contained summary of the theoretical background.
Consider the pricing of an arithmetic average fixed strike Asian put option in the Black-
Scholes model. Except for S0 = 100, δˆ = 0.01, and K = [95 100 105], the parameters of the
problem are group-specific and can be found in Table 1 overleaf.
1. Calculate the price of a discretely monitored option with initial value included in the
average, using weekly monitoring, ∆t = 1/50, by explicit finite difference scheme.
2. With the same parameter values, calculate the price of a forward start put option
whose averaging starts at the end of week 10.
3. Examine the convergence pattern for your scheme. Where appropriate, consider
extrapolation (Richardson’s or otherwise) of the results with respect to grid spacing.
Estimate the precision of your best result by exploiting the convergence pattern.
Using such estimate report prices that, in your opinion, are accurate to 10−3.
4. Compare your price with an explicitly calculated price of a suitable geometric aver-
age contract.
In your report, pay particular attention to four aspects of this assignment:
A) Transformation of the original problem by changing state variables and/or measure.
State clearly what transformations you have chosen to adopt and why.
B) Discretization errors: In grid schemes the price may oscillate with the number of
spatial steps. Describe the strategy you have chosen to circumvent such oscillation.
C) Curtailing the range: in grid-based methods one must choose a finite range for the
state variables in question. Justify your choice of range (by probabilistic arguments or by
devising an algorithm which determines the correct range adaptively).
D) Boundary conditions: An informed choice of boundary conditions is crucial for good
numerical performance. Justify your choice of the boundary condition by appealing to the
price of deep-in-the-money Asian call.
SMM306 Advanced Stochastics Coursework, ©2021 Alesˇ Cˇerny´ 2
Group No σ r T
1 0.2 0.01 3
2 0.3 0.06 1.5
3 0.2 0.08 3
4 0.4 0.1 1
5 0.2 0.12 3
6 0.6 0.02 1
7 0.3 0.04 2
8 0.4 0.05 2
9 0.3 0.06 2
10 0.5 0.1 1.5
11 0.3 0.08 2
12 0.6 0.04 1.5
13 0.3 0.1 2
14 0.4 0.02 1.5
Table 1: Model parameter specification